A Convergence Analysis on URV Refinement

TL;DR

This paper provides a new convergence proof for Stewart's URV refinement algorithm, improving understanding of its theoretical properties under weaker assumptions.

Contribution

It offers a novel convergence analysis for Stewart's URV refinement, extending previous results with less restrictive conditions.

Findings

Convergence of the refinement iteration is established under weaker assumptions.

The proof enhances theoretical understanding of the URV decomposition process.

Results support the algorithm's reliability in practical applications.

Abstract

Recently, Stewart gave an algorithm for computing a rank revealing URV decomposition of a rectangular matrix. His method makes use of a refinement iteration to achieve an improved estimate of the smallest singular value and its corresponding singular vectors of the matrix. Here, a new proof is given for the convergence of the refinement iteration. This analysis is carried out under slightly weaker assumptions than those of Mathias and Stewart.